Question

Use the graph to write a polynomial function of least degree

Answer 1

f(x) = -18x³ + 21x² + 10x - 8

Step-by-step explanation

As we can see, the x-intercepts are x = -2/3, x = 1/2 and x = 4/3. Thus, these are the zeros of the polynomial.

Given a polynomial with zeros x₁, x₂, and x₃, the polynomial's degree is 3 and it can be written with the factors,

`f(x)=a(x-x_1)(x-x_2)(x-x_3)`

Where a is a coefficient given by the location of the y-intercept.

With the zeros, we have the function,

`f(x)=a\mleft(x+(2)/(3)\mright)\mleft(x-(1)/(2)\mright)\mleft(x-(4)/(3)\mright)`

In the graph, the y-intercept is shown: (0, -8). This means that when x is 0, f(0) is -8. This is what we have to use to find the coefficient a,

`-8=a\mleft(0+(2)/(3)\mright)\mleft(0-(1)/(2)\mright)\mleft(0-(4)/(3)\mright)`

We have the product,

`-8=a\mleft((2)/(3)\mright)\mleft(-(1)/(2)\mright)\mleft(-(4)/(3)\mright)`

Solve the product,

`\begin{gathered} -8=a\mleft((2(-1)(-4))/(3\cdot2\cdot3)\mright) \\ -8=a\mleft((4)/(9)\mright) \end{gathered}`

To find a, multiply both sides by 9,

`\begin{gathered} -8\cdot9=a\cdot(4)/(9)\cdot9 \\ -72=4a \end{gathered}`

And divide both sides by 4,

`\begin{gathered} -(72)/(4)=(4a)/(4) \\ -18=a \end{gathered}`

Thus the coefficient a is -18, and the function is,

`f(x)=-18\mleft(x+(2)/(3)\mright)\mleft(x-(1)/(2)\mright)\mleft(x-(4)/(3)\mright)`

Now we have to write it in standard form. To do so, we have to multiply the factors to get a function in the form,

`f(x)=ax^3+bx^2+cx+d`

Apply the distributive property to the last two factors,

`\begin{gathered} f(x)=-18\mleft(x+(2)/(3)\mright)\mleft(x\cdot x-x\cdot(4)/(3)-x\cdot(1)/(2)+(1)/(2)\cdot(4)/(3)\mright) \\ \\ f(x)=-18\mleft(x+(2)/(3)\mright)\mleft(x^2-(11)/(6)x+(2)/(3)\mright) \end{gathered}`

Then do the same with the first factor,

`\begin{gathered} f(x)=-18\mleft(x^2\cdot x-(11)/(6)x\cdot x+(2)/(3)x+(2)/(3)x^2-(11)/(6)\cdot(2)/(3)x+(2)/(3)\cdot(2)/(3)\mright) \\ \\ f(x)=-18\mleft(x^3-(11)/(6)x^2+(2)/(3)x^2+(2)/(3)x-(11)/(9)x+(4)/(9)\mright) \end{gathered}`

Add like terms,

`f(x)=-18\mleft(x^3-(7)/(6)x^2-(5)/(9)x+(4)/(9)\mright)`

And finally, multiply each term by -18,

`\begin{gathered} f(x)=-18x^3+18\cdot(7)/(6)x^2+18\cdot(5)/(9)x-18\cdot(4)/(9) \\ f(x)=-18x^3+21x^2+10x-8 \end{gathered}`

Hence, the polynomial function whose graph is in the question is f(x) = -18x³ + 21x² + 10x - 8